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Comparative analysis of linear solvers for the Jacobian-free Newton–Krylov method for solving gas dynamics problems using an implicit scheme for the Discontinuous Galerkin Method
| Title | Comparative analysis of linear solvers for the Jacobian-free Newton–Krylov method for solving gas dynamics problems using an implicit scheme for the Discontinuous Galerkin Method |
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| Authors | R. V. Zhalnin1, A. D. Mulyugin1, V. V. Vdovin1 1National Research Mordovia State University |
| Annotation | A comparative study of the efficiency of linear solvers within the Jacobian-free Newton–Krylov (JFNK) framework for the numerical simulation of two-dimensional gas dynamics equations using the discontinuous Galerkin method is presented. The classical MINRES method and the modern induced dimension reduction method IDR(s) with various values of parameter s are considered. The study is performed on the example of the Kelvin—Helmholtz instability problem. The accuracy of the obtained numerical solutions, spectral characteristics of the flow, and computational performance of the algorithms are analyzed. All IDR(s) variants are shown to provide a quality of vortex structure resolution and energy spectra comparable to that of MINRES. Meanwhile, all methods from the IDR(s) family require significantly less wall-clock time per time step compared to MINRES, achieving a speedup factor from 1.5 to 2.5, depending on the dimension of the auxiliary subspace. The IDR(4) method demonstrates the best balance between convergence rate and computational cost, which allows us to recommend it as an efficient linear solver within the JFNK approach for unsteady computational fluid dynamics problems. |
| Keywords | Discontinuous Galerkin Method, Implicit Scheme, Jacobian-free Newton–Krylov Method, MINRES method, IDR(s) method |
| Citation | Zhalnin R. V., Mulyugin A. D., Vdovin V. V. ''Comparative analysis of linear solvers for the Jacobian-free Newton–Krylov method for solving gas dynamics problems using an implicit scheme for the Discontinuous Galerkin Method'' [Electronic resource]. Mathematical modeling, numerical methods, and software systems: Collection of materials from the 12th All‑Russian Scientific Youth School‑Seminar named after E. V. Voskresensky (Saransk, July 21–24, 2026). - pp. 77-82. Available at: https://conf.svmo.ru/files/2026/papers/paper78.pdf. - Date of access: 25.09.2026. |
© SVMO, National Research Mordovia State University, 2026
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